During a one-hour trip, a small boat travels 80.0 km north and then travels 60.0 km east.
What is the direction of the boat's average velocity for the one-hour trip?

Options

A. due east
B. `41.7^(@)` north of east
C. `53.1^(@)` north of east
D. `49.2^(@)` north of east

Answer

To find the direction of the boat's average velocity during its trip, we can follow these steps:

### Step 1: Understand the Displacement
The boat travels 80.0 km north and then 60.0 km east. We need to find the resultant displacement vector from the starting point to the final point.

### Step 2: Represent the Displacement Vectors
We can represent the displacements as vectors:
- The displacement vector for the northward journey can be represented as \( \vec{A} = 80.0 \, \text{km} \, \hat{j} \) (where \( \hat{j} \) is the unit vector in the north direction).
- The displacement vector for the eastward journey can be represented as \( \vec{B} = 60.0 \, \text{km} \, \hat{i} \) (where \( \hat{i} \) is the unit vector in the east direction).

### Step 3: Calculate the Resultant Displacement
The resultant displacement vector \( \vec{R} \) can be calculated as:
\[ \vec{R} = \vec{A} + \vec{B} = 80.0 \, \hat{j} + 60.0 \, \hat{i} \]

### Step 4: Calculate the Magnitude of the Resultant Displacement
To find the magnitude of the resultant displacement, we use the Pythagorean theorem:
\[ R = \sqrt{(80.0)^2 + (60.0)^2} \]
Calculating this gives:
\[ R = \sqrt{6400 + 3600} = \sqrt{10000} = 100.0 \, \text{km} \]

### Step 5: Determine the Direction of the Resultant Displacement
To find the angle \( \theta \) that the resultant displacement makes with the east direction, we can use the tangent function:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{80.0}{60.0} \]
Thus,
\[ \tan \theta = \frac{4}{3} \]
Now, we can find \( \theta \) using the arctangent function:
\[ \theta = \tan^{-1}\left(\frac{4}{3}\right) \]
Calculating this gives:
\[ \theta \approx 53.13^\circ \]

### Step 6: State the Direction of Average Velocity
The direction of the average velocity is therefore \( 53.13^\circ \) north of east.

### Final Answer
The direction of the boat's average velocity for the one-hour trip is approximately **53.1 degrees north of east**.

---